Theorem dfdm3 4883

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4703	. 2 |- dom A = { x | E. y x A y }
2		df-br 4060	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1629	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2323	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2228	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   <.cop 3646   class class class wbr 4059   dom cdm 4693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-br 4060  df-dm 4703

This theorem is referenced by:  csbdmg  4891